$\nabla ·\left(\nabla \times \mathbf{F}\right)$

$\= 0$

$\nabla \times \left(\nabla f\right)$

$\= \mathbf{0}$

$\nabla \left(f g\right)$

$\= f \nabla g\+g \nabla f$

$\nabla \left(\mathbf{F}·\mathbf{G}\right)$

$\= \left(\mathbf{F}·\nabla \right)\mathbf{G}\+\left(\mathbf{G}·\nabla \right)\mathbf{F}\+\mathbf{F}\times \left(\nabla \times \mathbf{G}\right)\+\mathbf{G}\times \left(\nabla \times \mathbf{F}\right)$

$\nabla ·\left(f \mathbf{F}\right)$

$\= f \nabla ·\mathbf{F}\+\mathbf{F}·\left(\nabla f\right)$

$\nabla ·\left(\mathbf{F}\times \mathbf{G}\right)$

$\= \mathbf{G}·\left(\nabla \times \mathbf{F}\right)-\mathbf{F}·\left(\nabla \times \mathbf{G}\right)$

$\nabla \times \left(f \mathbf{F}\right)$

$\= f \nabla \times \mathbf{F}\+\left(\nabla f\right)\times \mathbf{F}$

$\nabla \times \left(\mathbf{F}\times \mathbf{G}\right)$

$\= \left(\mathbf{G}·\nabla \right)\mathbf{F}-\left(\mathbf{F}·\nabla \right)\mathbf{G}\+\mathbf{F} \nabla ·\mathbf{G}-\mathbf{G} \nabla ·\mathbf{F}$

Table 9.4.1   Vector differential identities

Example 9.4.1

If $\mathbf{F}\=u\left(x\,y\,z\right) \mathbf{i}\+u\left(x\,y\,z\right) \mathbf{j}\+w\left(x\,y\,z\right) \mathbf{k}$, show that for sufficiently well-behaved $u\,v\,w$, the curl of F is solenoidal.

Example 9.4.2

If $f\=x y^2{z}^3$, show that $\nabla f$ is curl-free.

Example 9.4.3

Show that for sufficiently well-behaved functions $f\left(x\,y\,z\right)$, the gradient of $f$ is curl-free.

Example 9.4.4

If F is a sufficiently well-behaved vector field in cylindrical coordinates, show that $\nabla \times \mathbf{F}$ is solenoidal.

Example 9.4.5

If F is a sufficiently well-behaved vector field in spherical coordinates, show that $\nabla \times \mathbf{F}$ is solenoidal.

Example 9.4.6

Show that for sufficiently well-behaved functions $f\left(r\,\mathrm{\θ}\,z\right)$ in cylindrical coordinates, the gradient of $f$ is curl-free.

Example 9.4.7

Show that for sufficiently well-behaved functions $f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)$ in spherical coordinates, the gradient of $f$ is curl-free.

Example 9.4.8

For sufficiently well-behaved scalars $f\left(x\,y\,z\right)$ and $g\left(x\,y\,z\right)$, verify Identity 3 in Table 9.4.1.

Example 9.4.9

For a sufficiently well-behaved scalar $f\left(x\,y\,z\right)$ and vector field$\mathbf{F}\=u\left(x\,y\,z\right) \mathbf{i}\+u\left(x\,y\,z\right) \mathbf{j}\+w\left(x\,y\,z\right) \mathbf{k}$, verify Identity 5 in Table 9.4.1.

Example 9.4.10

For a sufficiently well-behaved scalar $f\left(x\,y\,z\right)$ and vector field$\mathbf{F}\=u\left(x\,y\,z\right) \mathbf{i}\+u\left(x\,y\,z\right) \mathbf{j}\+w\left(x\,y\,z\right) \mathbf{k}$, verify Identity 7 in Table 9.4.1.

Example 9.4.11

For sufficiently well-behaved vector fields $\mathbf{F}\=u \mathbf{i}\+v \mathbf{j}\+w \mathbf{k}$ and $\mathbf{G}\=a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}$, where $u\,v\,w$ and $a\,b\,c$ are functions of $x\,y\,z$, verify Identity 6 in Table 9.4.1.

Example 9.4.12

For sufficiently well-behaved vector fields $\mathbf{F}\=u \mathbf{i}\+v \mathbf{j}\+w \mathbf{k}$ and $\mathbf{G}\=a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}$, where $u\,v\,w$ and $a\,b\,c$ are functions of $x\,y\,z$, verify Identity 4 in Table 9.4.1.

Example 9.4.13

For sufficiently well-behaved vector fields $\mathbf{F}\=u \mathbf{i}\+v \mathbf{j}\+w \mathbf{k}$ and $\mathbf{G}\=a \mathbf{i}\+b \mathbf{j}\+c \mathbf{k}$, where $u\,v\,w$ and $a\,b\,c$ are functions of $x\,y\,z$, verify Identity 8 in Table 9.4.1.

Example 9.4.14

Working with sufficiently well-behaved quantities in polar coordinates, verify Identity 5 in Table 9.4.1 for $f\left(r\,\mathrm{\θ}\right)$ and $\mathbf{F}\=u\left(r\,\mathrm{\θ}\right) {\mathit{e}}_{\mathit{r}}\+v\left(r\,\mathrm{\θ}\right) {\mathit{e}}_{\mathbf{\θ}}$.

Example 9.4.15

Working with sufficiently well-behaved quantities in spherical coordinates, verify Identity 5 in Table 9.4.1 for $f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)$ and $\mathbf{F}\=u {\mathit{e}}_{\mathbf{\ρ}}\+v {\mathit{e}}_{\mathbf{\φ}}\+w {\mathit{e}}_{\mathbf{\θ}}$, where $u\,v\,w$ are functions of $\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}$.